Show as option not showing in calendar

All the devices sync with an email client server which hosts your calendar. If you have a Calendar on iCloud that will not sync with Google Calendar. Settings > Calendar > Default Calendar [Choose what you want]. What you add (Event) on iPhone Calendar is synced with default calendar unless you have chosen a specific calendar at the time of event creation.

To which email account are the calendar events synced with your iPhone? This is not clear in your post

See the iPhone and Mac Calendar Pics below

Which is your default calendar? Settings > Calendar > Default Calendar

Settings > Calendar > Sync > [All Events] - Watch the GIF below